Modeling Wireless Networks for Rate Control. David C. Ripplinger

Transcription

1 Modeling Wireless Networks for Rate Control David C. Ripplinger A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Sean Warnick, Chair Daniel Zappala Mark Clement Department of Computer Science Brigham Young University December 2011 Copyright c 2011 David C. Ripplinger All Rights Reserved

2 ABSTRACT Modeling Wireless Networks for Rate Control David C. Ripplinger Department of Computer Science, BYU Master of Science Congestion control algorithms for wireless networks are often designed based on a model of the wireless network and its corresponding network utility maximization (NUM) problem. The NUM problem is important to researchers and industry because the wireless medium is a scarce resource, and currently operating protocols such as often result in extremely unfair allocation of data rates. The NUM approach offers a systematic framework to build rate control protocols that guarantee fair, optimal rates. However, classical models used with the NUM approach do not incorporate partial carrier sensing and interference, which can lead to significantly suboptimal performance when actually deployed. We quantify the potential performance loss of the classical controllers by developing a new model for wireless networks, called the first-principles model, that accounts for partial carrier sensing and interference. The first-principles model reduces to the classical models precisely when these partial effects are ignored. Because the classical models can only describe a subset of the topologies described by the first-principles model, the score for the first-principles model gives an upper bound on the performance of the others. This gives us a systematic tool to determine when the classical controllers perform well and when they do not. We construct several representative topologies and report numerical results on the scores obtained by each controller and the first-principles optimal score. Keywords: wireless networks, rate control, modeling, partial carrier sensing, partial interference

3 ACKNOWLEDGMENTS Thanks go to Christopher Grant in the BYU Math Department for his help with developing the theorems and definitions for uniform random sets.

4 Table of Contents List of Figures vi List of Tables viii 1 Introduction 1 2 The Maximal Clique Model Constraints Objective Solution and Controller Design The Partial Interference Model Constraints Objective Solution and Controller Design Comparison to the Maximal Clique Model: Numerical Results Performance Metric Results The First-Principles Model A Simple Example Union of Uniform Random Sets Constraints Objective iv

5 5 Reduction of the First-Principles Model to the Classical Models Reduction to the Maximal Clique Model Reduction to the Partial Interference Model Solution to the First-Principles Optimization Problem Tightening the Upper Bound Implementation Performance of the Classical Controllers: Numerical Results Two-Link Topologies Three-Link Topologies Partial Interference Topologies Partially Dependent Interferers Chain Topology Mesh Topology Summary of Results Conclusion 59 A Branch and Bound Code 63 References 85 v

6 List of Figures 2.1 An example wireless network and its contention graph Topologies used to compare classical controllers Performance of classical controllers with I interferers on one link Performance of classical controllers with N contenders with one interferer Performance of classical controllers with I interferers and N contenders Example of the first-principles model, random overlap Example decision tree for the branch and bound method Two-link topologies Optimality CDFs of the classical controllers for the two-link topologies Infeasibility CDFs of the classical controllers for the two-link topologies Three-link topologies varied over interference Optimality CDFs for the three-link topologies varied over interference Three-link topologies varied over contention Optimality CDFs for the three-link topologies varied over contention Optimality CDFs for the partial interference topologies Maximal clique controller performance for partial interference topologies Partial interference controller performance for partial interference topologies Topologies with partially dependent interferers Optimality of classical models for partially dependent interferers Chain topology vi

7 7.14 Mesh topology vii

8 List of Tables 2.1 Notation used in the maximal clique model Notation used in the partial interference model Notation used in the first-principles model Parameters in the mesh topology viii

9 Chapter 1 Introduction Wireless networks are often used as a low-cost alternative to wired infrastructure, while also accommodating mobile users. The most prevalent medium access control (MAC) protocol used in wireless networks is defined in the IEEE standard. However, research has shown that when the wireless network is extended to multiple hops, the MAC is plagued with serious fairness and efficiency problems, sometimes completely starving one data flow in favor of another [1, 2, 3]. This is in part due to the fact that sharing resources in a wireless network is a fundamentally different problem than in a wired network, and requires some theoretical understanding to solve. As a result of these problems, rate allocation in wireless networks to achieve maximum network utilization and fairness has become a popular area of research. Seminal research in this area includes [4, 5] for wired networks, and has been readily extended to wireless networks in [6, 7, 8, 9, 10, 11, 12]. See [13] for a survey on this and other active research topics for wireless networks. Some research seeks to improve the capacity of a mesh network by introducing multiple radios operating on different frequencies, or by manipulating the transmission power of the radios in order to reduce contention [14, 15]. We view this work as complementary to ours. We limit the scope of our research area to designing transmission rate controllers for either communication flows or links, where algorithms manipulating routes, radio frequencies, antenna direction, or transmission power are already employed, and such algorithms are quasi-static with respect to the rate controller. 1

10 In this thesis, we will answer the following question: When do certain classical controllers behave poorly due to partial carrier sensing and interference effects? Congestion control algorithms are often designed based on a model of the wireless network and its corresponding network utility maximization(num) problem. In the network utility maximization (NUM) approach, an objective function for the network is defined, typically a sum of utility functions for each link s or flow s sending rate, where the form of the utility function defines a particular notion of fairness. Next, a set of constraints is used to model the unique characteristics of the wireless network, such as carrier sensing or interference constraints. The solution to this optimization problem will then yield a set of rates that maximize network utility for the links or flows. These rates can then be used as input to a rate controller that sits on top of (or inside) the MAC protocol, limiting the packet transmission rate for each flow or link. When the optimization problem is convex, it can often be translated into a distributed rate control algorithm, making it practical to deploy in a wireless network. Our focus is on the constraints used to model the wireless network, as this is the critical piece in the NUM approach. If the model is inaccurate, then the optimization problem may not yield an accurate or optimal solution. We limit our study to stationary, multi-hop wireless networks that use carrier sense multiple access(csma), such as the MAC. CSMA protocols work by each node listening to see if the medium is occupied before sending data, as opposed to working out a precise schedule with other nodes of which times to send. This broadly characterizes the most widely-used wireless networks in the field, often referred to as mesh networks. We present a first-principles model of wireless networks for the rate control or NUM problem. By first-principles, we mean that the most basic assumptions are made of how multihop wireless networks with CSMA operate. In this model, perceived times that the medium is occupiedarerepresented asarandomset. Ourmodelmayalsobeclassifiedasameasurement based model, as it takes as inputs the probabilities of links carrier sensing or interfering with each other. Such an approach is more realistic than physical layer modeling, such as the 2

11 various signal fading models with SINR thresholds, and has been used to model wireless networks for various purposes [16, 17, 18, 19, 20, 21, 11]. A combination of measurement based modeling and physical layer modeling is used in [18] to determine probabilities of carrier sensing and interference between pairs of nodes. Kashyap, et al [20] extends this idea to also model probabilities of carrier sensing and interference of groups of nodes in order to determine the capacity of a wireless link. We note specifically that their approach is very similar to ours, modeling the states (transmitting, deferring, idle) of a node as random sets. However, they also model specific aspects of the MAC as opposed to generic CSMA, and their model predicts an uncontrolled environment, where there is no rate controller other than the MAC. In the future, we hope to combine the qualities of ours and Kashyap s models to achieve further accuracy for rate control problems. We show that, under limiting conditions, our model reduces to previously proposed models in the literature. Specifically, with the assumption of binary, symmetric sensing, our model reduces to the common maximal clique model used by seminal research in this area [6, 10]. Likewise, with the assumption of no carrier sensing between interfering links, our model reduces to the partial interference model developed by Niculescu [21] and later employed in [12]. Because the first-principles model induces a non-convex NUM problem, it is too complex to deploy an efficient, distributed rate controller that solves it. However, we can solve it offline using a branch and bound method on several constructed network topologies that are representative of scenarios that often occur in real wireless networks. The objective function of our first-principles NUM problem can serve as a scoring mechanism for various rate inputs that any controller may compute, and the optimal solution will be an upper bound on the performance of the classical NUM-based rate controllers. We implement a branch and bound algorithm to solve the problem, and we compare its solution (the upper bound) on several representative topologies with the scores achieved by the classical NUM- 3

12 based controllers. In this way, we determine under what kinds of topologies partial carrier sensing and interference adversely affect the performance of the classical controllers. Some interesting scenarios were found in which the classical controllers performed as low as 33% of the optimal. The maximal clique controller was found to perform poorly in many small scenarios, especially those in which there was much partial (and little binary) carrier sensing or interference. Much fewer cases were found when the partial interference controller performed poorly. These were primarily limited to cases of significant partial carrier sensing (with little or no interference in the network), and of very heavy interference by several links that perfectly carrier sensed each other. We also computed the performance of the classical controllers on a chain topology and a mesh topology, which were larger and more typical of real networks than the toy topologies we explored. In both cases, the maximal clique controller performed above 86% and the partial interference controller above 97% of the optimal. We believe that these results are a strong indication that the partial interference model is good enough in most networks for rate control. Furthermore, we now have a method of detecting poor performance in a network, along with motifs that are easy to isolate as the culprits. 4

13 Chapter 2 The Maximal Clique Model The maximal clique model is the most widely used model for rate optimization in wireless networks [6, 10]. The notation used in this model is given in Table Constraints Figure 2.1 shows how a contention graph is inferred from a wireless network. This graph has a vertex representing each active link, and each edge signifies that two links contend, or cannot send at the same time. The maximal cliques are then identified. A clique is a subgraph such that each pair of its vertices share an edge. A maximal clique is a clique such that no other vertex in the parent graph could be included to form another clique. For each maximal clique j, its links sending rates s must sum to at most some clique capacity, which for our purposes will always be 1: s i 1, j C. (2.1) i L(j) s i Sending rate of link i. L Set of links. C Set of maximal cliques. L(j) Set of links in maximal clique j. C(i) Set of maximal cliques containing link i. Table 2.1: Notation used in the maximal clique model. 5

14 Figure 2.1: An example wireless network and its corresponding contention graph with the maximal cliques circled. Links 1, 2, and 3 contend because they share node B. Links 4 and 5 contend because sending nodes D and F are within carrier sensing range of each other. Assuming RTS/CTS is enabled, links 3 and 4 contend because sending node D is within carrier sensing range of receiving node C, which sends out CTS signals. This model infers that links in a maximal clique have sending rates summing to at most some clique capacity. 6

15 The constraints specific to Figure 2.1 are s 1 +s 2 +s 3 1, (2.2) s 3 +s 4 1, (2.3) s 4 +s 5 1. (2.4) Allowing non-unity values for clique capacities is a technique to account for less than ideal usage of the medium and indirect scheduling conflicts. The latter is demonstrated in [6] by showing a ring contention graph of 5 links, where (2.1) with clique capacities of 1 is inaccurate, predicting that each s i = 1/2. This allocation causes a scheduling conflict. If links 1 and 3 send during the first half of a block of time, and links 2 and 4 send during the second half, link 5 cannot send at all, since it contends with link 1 during the first half and link 4 during the second half. Measuring clique capacities less than one would allow the model to constrain the sending rates only to those such that a feasible schedule exists. One of the advantages of building the contention graph is that contention can be interpreted mathematically as a simple sum of rates being constrained, no matter the reason for contention. The most common reasons for contention are that two links share the same sending node or their sending nodes are within carrier sensing range of each other. Note, however, that any reason for two links contending can be appropriately modeled by the contention graph. For example, in Figure 2.1, links 3 and 4 contend because RTS/CTS (request-to-send/clear-to-send) is enabled, so that sending node D is within carrier sensing range of receiving node C. Node C will send out CTS signals to B, but D will overhear the CTS signals and defer the medium. 2.2 Objective The objective function in the optimization problem must yield high throughput while also providing fairness between flows. It is well known that these two goals are conflicting for 7

16 wireless networks to provide fairness, some flows may need to sacrifice throughput. A systematic way of achieving a balance between the two is by constructing the objective function as a sum of rate utility functions, which are concave and strictly increasing. The type of fairness is determined by the utility function s degree of concavity. Several notions of fairness and their corresponding utility functions are well established in the literature [22, 23]. The two most common notions of fairness are max-min fairness and proportional fairness. Max-min fairness can be understood as the rates achieved by increasing all the sending rates from zero simultaneously until a constraint is active. Then continue to increase the sending rates for those links that still have not been affected by some active constraint until the next constraint becomes active, repeating the process until all links have been maximized. These rates are achieved in the NUM problem by choosing the utility function U(s i ) = s α i /α, α, (2.5) which can be approximated by choosing a large α. Proportional fairness seeks to achieve equal rates among links, but it allows for one link to lower its rate somewhat if it allows for multiple other links to then increase their rates a significant amount. It is achieved in the NUM problem by choosing the utility function U(s i ) = lns i. (2.6) For this and subsequent models, we will only consider proportional fairness for two reasons. First, max-min fairness often results in significant lost overall bandwidth in the network. Second, while there exists a distributed solution to the maximal clique NUM problem for any notion of fairness, the logarithms in proportional fairness allow decoupling of variables in subsequent models, so that a distributed solution exists for the partial interfer- 8

17 ence NUM problem, and a tighter upper bound on the first-principles NUM problem can be calculated. For our purposes then, the objective function for the maximal clique model is f(s) = i L lns i. (2.7) 2.3 Solution and Controller Design Here we show how the maximal clique NUM problem can be solved distributively and the high-level considerations in deploying a controller to implement it. The development is adapted from [10]. It is implicit in this and all problems in this thesis that rates are constrained to be non-negative. The problem is maximize subject to lns i i L i L(j) s i 1, j C. (2.8) To solve it, we write the Lagrangian, which is constructed by adding to the objective function a linear combination of the constraint functions: L(s,λ) = lns i + λ j 1 i L j C i L(j) s i. (2.9) Each λ j can be thought of as a price for maximal clique j, or the per-unit cost of breaking constraint j. They are also called the Lagrange multipliers. Note that maximizing the Lagrangian over s can be thought of as a relaxation of the original problem. Instead of breaking the constraints being infinitely bad, one only incurs a finite cost. Before moving on, we first rearrange the terms in (2.9) to get L(s,λ) = i L (lns i Λ i s i )+ j C λ j, (2.10) 9

18 where Λ i = j C(i) λ j is the sum of prices for each clique that link i belongs to. Next we construct the dual function, which is simply the Lagrangian maximized over s: D(λ) = max s (lns i Λ i s i )+ λ j. (2.11) j C i L The max can be brought inside the sum to get D(λ) = max(lns i Λ i s i )+ λ j. (2.12) s i i L j C The function ρ(s i,λ) = lns i Λ i s i (2.13) being maximized over s i is concave and not monotonic, so that a unique maximizer s i (λ) is guaranteed. We can solve for it analytically by taking the derivative and setting it equal to zero: Substituting this into (2.12) results in s i (λ) = Λ 1 i. (2.14) D(λ) = i L ( lnλ 1 i 1 ) + j C λ j. (2.15) The dual function, for any non-negative λ, gives an upper bound on the original objective function in (2.8). In order to tighten this upper bound, we solve the dual problem, which is to minimize the dual function over λ: minimize i L ( lnλ 1 i 1 ) + j Cλ j subject to λ j 0, j C. (2.16) When the optimal score for both the primal problem and the dual problem are equal, it is said that they have strong duality. For convex problems, strong duality holds when Slater s 10

19 condition is met, that is, there exists a feasible point in the relative interior of the domain. Slater s condition holds for all problems in this thesis. We thus have strong duality for (2.8) and (2.16), and the corresponding optimal sending rates are s = s(λ ), where λ are the optimal prices. The dual problem is solved using the gradient descent method. From Danskin s Theorem [24], we know that D = 1 λ j i L(j) s i. (2.17) Using a step size γ in the negative direction of the gradient gives the algorithm λ j (k +1) = max 0, λ j(k) γ 1 s i (λ(k)). (2.18) i L(j) Each link in a maximal clique can share with each other their current values of s i, which is only local information, and then compute the next λ j. The convergence of the algorithm is well established in the literature, even when it is asynchronous [5]. There are some difficulties with actually implementing this controller. It is well known that discovering all maximal cliques in a graph is NP-hard, so it would be necessary to approximate the set of maximal cliques. Also, if non-unity clique capacities are used in the model in order to make it more accurate, these capacities would have to be remeasured each time the optimization problem would need to be solved. Finally, the gradient descent method is frequently subject to slow convergence depending on the particular problem and choice of step size, in comparison to Newton-based methods. An implementation was deployed on a real network in [10], which had convergence rates on the order of tens of seconds. 11

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21 Chapter 3 The Partial Interference Model The partial interference model supplements the maximal clique model with constraints on the receiving rates [12]. The model is based on an empirical study of carrier sensing and interference in a wireless mesh network [21]. The notation used in this model is given in Table Constraints In the partial interference model, an interfering node may corrupt a fraction of the packets received at a remote node. Partial interference is not represented in the contention graph, but is instead represented in a directional, weighted interference map or matrix, and is incorporated as a constraint on receiving rates. To model partial interference accurately, we separate contention constraints from interference constraints. Contention is represented as an undirected edge between two vertices (links), and interference is modeled as a direcs i Sending rate of link i. r i Receiving rate of link i. d i Delivery ratio of link i. a ij Receiving interference probability of link j interfering with link i. L Set of links. L i Set of all links in L except i. C Set of maximal cliques. C(i) Set of maximal cliques containing link i. L(j) Set of links in maximal clique j. Table 3.1: Notation used in the partial interference model. 13

22 tional, weighted edge from the interfering link to the receiving link that is affected by the interference. The constraint on each receiving rate is r i = d i s i (1 a ij s j ), i L, (3.1) j L i wherethedeliveryratiod i impliesinherentlossoverthelink. InFigure2.1,assumeRTS/CTS is disabled so that node D can corrupt some packets received at C. The contention graph of Figure 2.1 would then be modified by replacing the edge between links 3 and 4 with a directional edge with weight a 34. Then (2.3) is dropped and in its stead we have r 3 = d 3 s 3 (1 a 34 s 4 ). (3.2) The partial interference model is less conservative than the maximal clique model because more links are assumed to transmit concurrently. For example, consider links 3 and 4 in Figure 2.1, and suppose link 4 corrupts 40% of packets received at link 3. If both links transmit at the clique capacity 1, then the sum of their effective receiving rates becomes 1+(1 0.4) = 1.6. In the maximal clique model, these links cannot transmit at the same time because they are in the same clique, resulting in a total effective receiving rate of 1. The partial interference model thus allows for significantly higher utilization of the network in certain scenarios. It is important to recognize that even complete interference cannot be accurately modeled as contention. That is, a link j will not become a contender to a remote link i even if the interference factor a ij = 1. Consider again the relationship between links 3 and 4 in Figure 2.1. Suppose a 34 = 1. If interference is modeled as contention, then both links will transmit at a rate of 0.5. However, the total effective receiving rate will be r 2 +r 3 = (0.5)(0.5)+0.5 = With the partial interference model, it is easy to see that link 3 should send at the full rate regardless of link 4 s rate. If link 4 continues to send at a 14

23 rate of 0.5, then the total effective receiving rate will be (1)(0.5)+0.5 = 1. Thus the partial interference model will result in higher utility. 3.2 Objective This leads to modifying the objective function. Note that we now care about receiving rates instead of sending rates. Therefore, the objective function for the partial interference model is f(r) = lnr i. (3.3) i L Here we also see why choosing proportional fairness is helpful mathematically. If r i in the objective function is replaced by (3.1), each term in the product converts to a sum of logarithms, so that each term in the summation is dependent on only one variable in s. 3.3 Solution and Controller Design The partial interference NUM problem can be solved distributively almost identically to the method for the maximal clique NUM problem. The development is adapted from [12]. Noting that delivery ratios d only add a constant to the objective function (3.3) when written as a function of s, we will simply assume from this point on that d i = 1 for all i. Rearranging the log terms on each s i in the objective, the primal problem is maximize subject to ( lns i + i L i L(j) ) ln(1 a ji s i ) j L i s i 1, j C. (3.4) Just as with the maximal clique NUM problem, we solve(3.4) by constructing the Lagrangian: L(s,λ) = ( lns i + ) ln(1 a ji s i ) + λ j 1 i L j L i j C i L(j) s i. (3.5) 15

24 We again rearrange the terms to get L(s,λ) = ) (lns i + j Li ln(1 a ji s i ) Λ i s i + λ j, (3.6) i L j C where Λ i = j C(i) λ j as before. We next construct the dual function: D(λ) = max s ρ(s i,λ)+ λ j (3.7) j C i L = maxρ(s i,λ)+ λ j, (3.8) s i i L j C where ρ(s i,λ) = lns i Λ i s i + j L i ln(1 a ji s i ). (3.9) Note that this development only differs from the maximal clique model s development by changing ρ(s i,λ) from (2.13) to (3.9), so that s i (λ) = argmax s i ρ(s i,λ) (3.10) produces a different answer by incorporating a cost for each time link i interferes with another link. It is difficult to solve (3.10) analytically, but it is easy to do numerically, for example with Newton s method. The dual problem is minimize i Lρ( s i (λ),λ)+ j C subject to λ j 0, j C. λ j (3.11) Again, we use (2.18) to solve the dual problem, and the optimal rates are given by s = s(λ ), where λ are the optimal prices. 16

25 3.4 Comparison to the Maximal Clique Model: Numerical Results We seek to determine in what situations the partial interference controller outperforms the maximal clique controller, and by how much. We use MATLAB to numerically compute solutions to the rate optimization problem for several different wireless networks. We use network topologies that represent basic situations these can be thought of as building blocks out of which larger topologies can be formed. We introduce three strains of the maximal clique controller that we compare with the partial interference controller. The interference-as-contention (IC) controller replaces any interference mappings with contention, no matter how small the interference factor a. The interference-ignored (II) controller simply ignores any interference mappings and models only contention. The adaptive contention (AC) controller follows the IC controller or the II controller, depending on which controller has higher performance. Thus the AC controller gives the maximal clique model the benefit of the doubt it ignores interference when this provides good performance and models it as contention otherwise Performance Metric To compare these different controllers, we define a performance metric that is based on the objective function of the partial interference model, using receiving rates. To make the performance metric intuitive, we use the geometric mean of the receiving rates, as this is a monotonic transformation of the partial interference objective function: P(r) = e f(r)/ L (3.12) The comparison should be made between the performance observed with receiving rates r derived from the partial interference controller and the receiving rates r actually obtainedbytheother controllerfromitssending ratess, according tothepartialinterference constraint on receiving rates. In other words, one of the maximal clique controllers comes up 17

26 with some sending rates s that it considers optimal. We take that s and substitute it into (3.1) to get r. Then we substitute both r and the partial interference controller s solution r into (3.12) to get their respective scores P. For ease of interpretation, we consider the ratio R of performances P, that is, R = P(r )/P(r ). (3.13) Thus, the comparison will simply read that the partial interference controller outperforms the maximal clique controller R times Results We consider three generic network topologies, shown in Figure 3.1 (the two topologies shown and a hybrid), and plot R for each topology and for each strain of the maximal clique controller being compared with the partial interference controller. Each topology is represented in the figures as a combined contention graph and interference map, where solid lines denote contention and dashed arrows denote the link at the arrow s tail interfering with the link at the arrow s head. In all cases, the IC controller never does as well as the partial interference controller because modeling interference as contention is too conservative. For low values of interference, it is better to let links send at faster rates and suffer some packet loss. At high values of interference, it is better to have the interfered link send at a faster rate than the interferer, to provide better throughput and fairness. However, modeling interference as contention is often better than ignoring it when interference is high. Thus in most cases, the hybrid AC controller follows the II controller for low values of interference and follows the IC controller for high values of interference. Figure 3.1(a) shows the first topology, where I links interfere with a single link with a common interference factor a, but do not interfere with each other. Figures 3.2(a), 3.2(b), 18

27 0 a 1 2 N 1 Clique 2 a 0 I (a) I links interfering with a single link. (b) N contenders in a clique with one interferer. Figure 3.1: Topologies used to compare the performance of the partial interference controller and the maximal clique controller R 1.4 R 1.4 R I 0 0 a I 0 0 a I 0 0 a (a) Ratio R of performance between the partial interference controller and the IC controller. (b) Ratio R of performance between the partial interference controller and the II controller. The dotted line marks where R begins to be greater than one. (c) RatioRofperformancebetween the partial interference controller and the AC controller. The dotted line marks where R begins to be greater than one. Figure 3.2: Numerical results for the performance of classical controllers with I interferers on one link. and 3.2(c) plot R for this topology for the partial interference controller against the IC, II, and AC controllers, respectively. The dotted lines show where R begins to be greater than one. Interestingly, the partial interference controller and the II controller perform exactly the same for values of a below This is because, for low values of a, the cost of interference is offset by the gain of the interferer sending at full capacity. Thus, both the partial interference controller and the II controller calculate sending rates at full capacity for each link. For larger values of a and I, the partial interference controller outperforms the maximal clique controllers more than 1.5 times. 19

28 R R 3 2 R N 0 0 a N 0 0 a N 0 0 a (a) Ratio R of performance between the partial interference controller and the IC controller. (b) Ratio R of performance between the partial interference controller and the II controller. The dotted line marks where R begins to be greater than one. (c) RatioRofperformancebetween the partial interference controller and the AC controller. The dotted line marks where R begins to be greater than one. Figure 3.3: Numerical results for the performance of classical controllers with N contenders with one interferer. Figure 3.1(b) shows the second topology, where a single link has interference factor a on N links that contend in a single clique. Figures 3.3(a), 3.3(b), and 3.3(c) plot R for this topology for the partial interference controller against the IC, II, and AC controllers, respectively. The dotted lines show where R begins to be greater than one. The partial interference controller starts performing better than the II controller at much lower values of awhenn islarge. Thisisduetothefactthatthecontendinglinksalreadyhavesmallratesas a consequence of sharing the medium. Utilities are lowered much more by interference when sending rates are small. Thus, even for low values of a, the partial interference controller does not calculate sending rates at full capacity. However, for higher values of a and N, the partial interference controller outperforms the IC controller only about 1.1 times. The AC controller consequently does relatively well across all values. To demonstrate the worth of the partial interference model, we consider a topology combining features of the first two, where I links have a fixed interference factor a = 0.4 on N links that contend in a single clique. Figures 3.4(a), 3.4(b), and 3.4(c) plot R for this topology for the partial interference controller against the IC, II, and AC controllers, respectively. Experimental results in [21] show that it is typical for interference factors to 20

29 2 3 2 R R R N I N I N I (a) Ratio R of performance between the partial interference controller and the IC controller. (b) Ratio R of performance between the partial interference controller and the II controller. (c) RatioRofperformancebetween the partial interference controller and the AC controller. Figure 3.4: Numerical results for the performance of classical controllers with I interferers and N contenders, with a = 0.4. range anywhere between zero and one in a real network, with usually at least one interferer on a link having a factor of at least a = 0.8, so choosing a = 0.4 in this topology is a reasonable comparison. The combined effect of several interferers and several contenders causes the partial interference controller to perform significantly better than the maximal clique controllers. 21

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31 Chapter 4 The First-Principles Model The design of the maximal clique and partial interference models (which we will refer to as the classical models) raises some questions. How do we know that mutually contending links (maximal cliques) implies that their rates must sum to at most 1 (or the clique capacity)? How can the maximal clique model be logically extended to consider the case of partial carrier sensing, where there is a continuous range of probabilities that nodes can sense each other, and the notion of cliques is immediately destroyed? Why, in the partial interference model, is the effect of each interfering link multiplicative? We seek to answer these questions by developing a model from a more theoretical standpoint by using random sets to represent observed times the medium is occupied. The notation used in the first-principles model is given in Table 4.1. The following elementary assumptions are made: Discretization of time. Time is divided into large blocks of time that are further divided into equally sized slots. During each time slot, each link is either sends or doesn t. Partial carrier sensing. Due to carrier sensing, there exists a fixed probability c ij that if link j is sending during a time slot, then the slot is not available for link i to send, when all other links are not sending. Partial receiving interference. There exists a fixed probability a ij that if links i and j send during a time slot, then link i does not successfully receive the data, when all other links are not sending. Also, due to inherent loss, there exists a fixed probability 23

32 s i Sending rate of link i. r i Receiving rate of link i. d i Delivery ratio of link i. a ij Receiving interference probability of link j interfering with link i. c ij Carrier sensing probability of link j being sensed by link i. S i Effective sending rate of all other links as observed by link i. R i Effective (receiving) interference rate at link i due to all other links. K i Set of links that contend with link i. p Number of elements in set p. p 1 \p 2 Set of elements in p 1 but not in p 2. P(p) Set of all subsets of p except the empty set. P z (p) Set of all subsets of p with p = z. f i (p) Sending rate product of links in p at link i. f i(p) Interference rate product of links in p at link i. g i (p) Free space term of links in p at link i. h(p) Independence of links in p. φ i (p) Transparency of link i to links in p. Table 4.1: Notation used in the first-principles model. d i that if link i sends data during a time slot, then it will be successfully received, when no other link is sending. Uniform random selection. For each time block T, each link has a set F T of available time slots in which to send, and a set X F when it does send. Each t F has an equal probability of being in X. Negligible indirect scheduling. Using the notation from the above assumption, if link i carrier senses links j and k sending during X j,x k T, respectively, then dependencies of X j X k on the sending times of any link l i,j,k is negligible. When considering the effective rate of links j and k as perceived by link i, realistically there may be some other link l (or even a set of other links) that cause the rates of j and k to overlap more or less than usual, which could in turn affect how much link i can send. The last assumption simply states that these effects are negligible. We recognize that it could 24

33 Contention graph Overlapping rates Figure 4.1: The contention graph of an example network, where one link in the middle contends with two outer links. If links 1 and 3 have rates of 1/2, and send at random times, then their effective rate or union will be 3/4 on average. have some impact on the accuracy of the model, but fine tuning the model in this way will not be addressed in this thesis. 4.1 A Simple Example Here we present a simple example to illustrate the design concepts behind the first-principles model. Consider the contention graph in Figure 4.1. Let us assume that at time k the rates are set at s 1 [k] = s 3 [k] = 1/2 and s 2 [k] = 0. What is the effective rate S 2 of links 1 and 3, as measured by link 2? The maximal clique model (2.1) suggests that S 2 = max(s 1,s 3 ) = 1/2. This implies that signals s 1 and s 3 overlap perfectly. If they didn t overlap at all, then S 2 = s 1 + s 3 = 1. In reality, there is some random overlap between the two. If we assume that the slots within the time block k during which 1 and 3 send are chosen at random, then, on average, half of the slots chosen by 1 will also be chosen by 3. Thus they overlap 1/4 of the total time, and the effective rate is S 2 = 3/4. In general, the effective rate S 2 actually depends on how much link 2 is sending, since it limits the available space over which links 1 and 3 can choose random sending times. It should be easy to see that as s 2 increases, s 1 and s 3 will overlap more, which lowers S 2. The 25

34 resulting formula is S 2 [k] = s 1 [k]+s 3 [k] s 1[k]s 3 [k] 1 s 2 [k]. (4.1) If we assume some kind of delay of link 2 affecting how much the others overlap, the dynamic system would behave according to s 2 [k +τ] 1 S 2 [k] (4.2) for some delay τ. The equilibrium constraint is s 2 +S 2 1. (4.3) 4.2 Union of Uniform Random Sets A link s sending rate is restricted by how much it senses that others are sending, in other words, their effective rate. Our goal is to find a formula for this effective rate by calculating the union of each random set representing a link s sending times. We begin by defining a uniform random set: Definition 1. Let T and F T be finite sets. Also let ξ t : Ω {0,1} for all t T be a collection of i.i.d. random variables on the probability space (Ω,A,P). Then X : Ω 2 T, where X(ω) = {t F : ξ t (ω) = 1}, (4.4) is a uniform random set on F. Moreover, if F is a random set, then X(ω) = {t F(ω) : ξ t (ω) = 1} (4.5) also defines a uniform random set on F. The set F is called the parent of X. 26

35 The deterministic set T can be considered as the entire time block. If link i sends at rate s i, link j hears c ji s i of T being occupied. Thus the c ij s j that link i hears must be a random set chosen from a subset of T, namely during which j did not hear i. For this reason we include in the definition of a uniform random set a parent set, or free space F, to which it is restricted. Now consider another link k adding to the effective rate that i senses. The effective rates of j and k by themselves overlap in the total effective rate a certain amount depending on how much they sense each other. In this sense, the above definition still does not fully explain the interaction of uniform random sets. We thus define the independence of uniform random sets: Definition 2. Let X be a collection of uniform random sets with parents F, enumerated by N = {1,...,n}. Then their independence is h(n) = E t F [ where E denotes the expected value. ] Pr(t X) i N Pr(t X, (4.6) i F) Let denote the expected size of a random set. Averaging over all t in F, we have from (4.6) that Pr(t X) = h(n) i NPr(t X i F), X F = h(n) i N X i F. F But X i F is found by multiplying X i by the probability that an element in F i is also in F: X i F = F F i X i 27

36 so that X = h(n) F i N X i F i. (4.7) We can now invoke the inclusion-exclusion principle X = ( 1) p 1 X X X p P(X) X p (4.8) to find the size of the union of many uniform random sets. Theorem 1. Let X be a collection of uniform random sets with parents F, enumerated by N = {1,...,n}. Then X = p P(N)( 1) p 1( i p ) i p F i X i i p F h(p). (4.9) i Proof. The result follows from (4.7) and (4.8). 4.3 Constraints The sending constraint is given by s i +S i 1, i L, (4.10) where S i = p P(L i ) ( 1) p 1 f i (p)g i (p)h(p), (4.11) f i (p) = j p c ij s j, (4.12) g i (p) = φ i(p) j p φ i(j), (4.13) 28

37 φ i (p) = 1 s i ( 1) p 1 c ji, (4.14) j p p P(p) and the independence is given by h(p) = (1 c ij c ji +c ij c ji ). (4.15) {i,j} P 2 (p) The receiving constraint is given by r i = d i (1 R i )s i, (4.16) where R i = p P(L i ) ( 1) p 1 f i (p)h(p) (4.17) and f i (p) = j pa ij s j. (4.18) Note that (4.15) is an approximation of (4.6). If one link carrier senses another link completely (c ij = 1) then their random sets do not intersect. If any two random sets in p do not intersect, then the intersection of p is empty, which means that h(p) should equal zero. Only when all random sets are independent should it equal one. The formulas for S i and R i follow immediately from Theorem 1. In the case of R i, since there is no intermediary correlating link for the interferers, F is the entire space with size 1. However, for S i, we need to derive g i (p), which corresponds to the term of free spaces F in the theorem. Link j observes a free space F j of size 1 c ji s i, in which link i observes an occupied space X j of size c ij s j. The free space F j consists of a portion during which link i is not sending, denoted Γ, and a portion during which link i is sending but not heard by j, denoted 29

38 Ψ j. Their sizes are given by Γ = 1 s i and Ψ j = (1 c ji )s i. Let F(p) = j p F j and Ψ(p) = j p Ψ j. Then, since every Ψ j is an independent uniform random set within a common space of size s i, we have by way of (4.7), Thus Ψ(p) = j p Ψ j s p 1 i = s i (1 c ji ). j p φ i (p) := F(p) = Γ + Ψ(p) = 1 s i +s i (1 c ji ) j p = 1 s i ( 1) p 1 c ji. j p p P(p) We call this function the transparency of i to p, and use it to simplify the notation in g i (p). As a side note, it is straightforward to incorporate flow constraints into the model as well, by introducing mappings t( ) from the hop number in the flow to the index of the link considered. Then s t(m) r t(m 1) (4.19) for each hop m and each flow mapping t ensures that no subsequent hop sends more than it receives. 30

39 4.4 Objective The objective function for the first-principles model is again the sum of the log of receiving rates, as in (3.3) for the partial interference model. 31

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41 Chapter 5 Reduction of the First-Principles Model to the Classical Models We must prove that the first-principles model reduces to the classical models. This is necessary to guarantee that the more general first-principles model can yield the exact same constraints as the classical models when there is no partial carrier sensing or interference, and thus perform at least as well as these models. 5.1 Reduction to the Maximal Clique Model The first-principles model reduces to the maximal clique model when carrier sensing is binary and symmetric. In other words, c ij = c ji {0,1}. We prove this by showing that the set of feasible sending rates in one model is equivalent to the feasible set in the other model. We first formally define the two sets of feasible rates based on each model: Definition 3. Given a contention graph (L,C), the set S 1 consists of all vectors of sending rates s that satisfy (2.1). Definition 4. Given a contention graph (L,C), the set S 2 consists of all vectors of sending rates s that satisfy (4.10), where S i = p P(K i ) ( 1 1 s i ) p 1 h(p) j p s j and 0, i,j p : i K j, h(p) = 1, otherwise. 33

42 Theorem 2. Given a contention graph (L,C), S 1 = S 2. Proof. First, note that S 1 is equivalent to the set of s satisfying s i +max{s i (j) : j C(i)} 1, i L, where S i (j) = l L(j)\i s l. To show that S 1 = S 2, it suffices to show that, for any i L, the constraint boundary is equivalent in S 1 and S 2. Thus, letting s i = 1 max{s i (j) : j C(i)}, we seek to show that S i = max{s i (j) : j C(i)}. Without loss of generality, let i = 0 and K 0 = {1,...,n}, where L(1) = {1,...,m} is the most constraining maximal clique and S 0 (1) = m j=1 s j. Then S 0 = ( ) p 1 1 h(p) s j S 0 (1) p P(K 0 ) j p n = ( ) z 1 1 h(p) s j S z=1 0 (1) p P z(k 0 ) j p n n ( ) z 1 1 = s j + )h(p) s j S j=1 z=2 0 (1) p P z(k 0 j p n n ( ) z 1 1 = S 0 (1)+ s j + )h(p) S j=m+1 z=2 0 (1) p P z(k 0 j p s j. We therefore must determine that n j=m+1 s j + n z=2 ( ) z 1 1 )h(p) S 0 (1) p P z(k 0 j p s j = 0. (5.1) 34

43 Note that h(p) = 0 for any p that has at least two elements from L(1)\{0}, so that p P z(k 0 )h(p) j p s j = S 0 (1) p P z 1 (K 0 \L(1))h(p) j p s j + p P z(k 0 \L(1))h(p) j p This is substituted into the argument of the second sum of (5.1) to obtain ( ) z 1 1 )h(p) s j = S 0 (1) p P z(k 0 j p ( 1)z 2 S 0 (1) z 2 p P z 1 (K 0 \L(1))h(p) j p s j + ( 1)z 1 S 0 (1) z 1 p P z(k 0 \L(1))h(p) j p s j. s j. The second term above for some z cancels with the first term in the corresponding z + 1 equation, and the first term for z = 2 cancels with n j=m+1 s j. We need only check that the second term for z = n goes to zero. By inspection, p P n(k 0 \L(1))h(p) j p s j = 0, because K 0 \L(1) < n. Thus, S 0 = S 0 (1) and S 1 = S Reduction to the Partial Interference Model We now show that when interferers of link i do not contend with each other, (4.16) and (4.17) from the first-principles model reduce to (3.1) in the partial interference model. To do this, we present a simple arithmetical theorem: Theorem 3. For some set L of indices, (1 x j ) = 1 x j. (5.2) j L p P(L)( 1) p 1 j p Proof. Without loss of generality, let L = {1,...,n}. We prove by induction. The base case n = 1 holds trivially. Assuming (5.2) holds for n, we need to show that it holds for n+1. 35

44 Defining N = L\(n+1), (1 x j ) j ) = (1 x n+1 j L(1 x ) j N = (1 x n+1 ) 1 p P(N)( 1) p 1 j p = 1 x j x n+1 + p P(N)( 1) p 1 j p = 1 x j. p P(L)( 1) p 1 j p x j ( 1) p 1 x n+1 p P(N) j p x j Applying this result to (3.1) is straightforward, replacing x j with a ij s j. Since under the limiting condition of no contending interferers we have h(p) = 1, we see that (4.16) does indeed reduce to (3.1). 36